2009-09-18

Ajay Ramadoss, Cornell University

A Differential form computing the Lefschetz number of a differential operator.

Abstract: Let $X$ be a compact complex manifold. The $i$-th Hochschild homology of the sheaf ${\mathcal D}_X$ of differential operators on $X$ (in a suitable sense) is known to be isomorphic to the $2n-i$-th cohomology of $X$ with complex coefficients. We shall discuss a construction of M. Engeli and G. Felder of an "explicit" quasi-isomorphism of complexes of sheaves between the complex of (completed) Hochschild chains of $D_X$ and the (smooth) De-Rham complex of $X$. As a consequence, for any global holomorphic operator $D$ on $X$, one can obtain an "explicit" $2n$-form whose integral over $X$ is the Lefschetz number of $D$.